SPA+RPA approach to canonical and grandcanonical treatments of nuclear level densities

Using an exactly solvable pairing model Hamiltonian in the static path approximation together with small-amplitude quantal fluctuation corrections in random phase approximation (SPA+RPA), we have analyzed the behaviour of canonical (number projected) and grandcanonical treatments of nuclear level densities as a function of temperature and number of particles. For small particle numbers at a low temperature, we find that though the grandcanonical partition function in SPA+RPA approach is quite close to its exact value, the small errors in its estimation causes significant suppression of level density obtained using number projected partition function. The results are also compared with the smoothed out exact values of level density. Within this model study, it appears that due to saddle point approximation to multiple Laplace-back transform, the grandcanonical treatment of level density at low temperature may be reliable only for relatively large number of particles. 1 E-Mail: bijay or [email protected] 1 The familiar Bethe formula[1, 2] for the level density, because of its simplicity, has been widely used to perform the statistical analysis of nuclear reactions. This level density formula takes a simple form due to (i) the connection of grandcanonical partition function with microcanonical partition function (level density) by a saddle point approximation in the evaluation of the traces over the states[3] and (ii) the grandcanonical partition function itself is approximated using independent single particle spectrum which is further assumed to be equidistant. Recently, a more realistic value of the level density[4]-[6] in saddle point approximation is obtained using a grandcanonical partition function in static path approximation[7, 8] which accounts for the large-amplitude thermal fluctuations. The method of saddle point approximation can be generalized in order to treat the level densities that are characterized by a set of quantum numbers, provided, these quantum numbers are composed additively of contributions from the single particle states (e.g. excitation energy (E), number of protons (Np) or neutrons (Nn), angular momentum, parity etc.). For instance, consider the system consisting of one kind of particles(protons or neutrons). The exact level density,